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Seznam integrálů hyperbolických funkcí

Z Multimediaexpo.cz

Toto je seznam integrálů (primitivních funkcí) hyperbolických funkcí.

\(\int\sinh cx\,\mathrm{d}x = \frac{1}{c}\cosh cx\)
\(\int\cosh cx\,\mathrm{d}x = \frac{1}{c}\sinh cx\)
\(\int\sinh^2 cx\,\mathrm{d}x = \frac{1}{4c}\sinh 2cx - \frac{x}{2}\)
\(\int\cosh^2 cx\,\mathrm{d}x = \frac{1}{4c}\sinh 2cx + \frac{x}{2}\)
\(\int\sinh^n cx\,\mathrm{d}x = \frac{1}{cn}\sinh^{n-1} cx\cosh cx - \frac{n-1}{n}\int\sinh^{n-2} cx\,\mathrm{d}x \qquad\mbox{(pro }n>0\mbox{)}\)
také: \(\int\sinh^n cx\,\mathrm{d}x = \frac{1}{c(n+1)}\sinh^{n+1} cx\cosh cx - \frac{n+2}{n+1}\int\sinh^{n+2}cx\,\mathrm{d}x \qquad\mbox{(pro }n<0\mbox{, }n\neq -1\mbox{)}\)
\(\int\cosh^n cx\,\mathrm{d}x = \frac{1}{cn}\sinh cx\cosh^{n-1} cx + \frac{n-1}{n}\int\cosh^{n-2} cx\,\mathrm{d}x \qquad\mbox{(pro }n>0\mbox{)}\)
také: \(\int\cosh^n cx\,\mathrm{d}x = -\frac{1}{c(n+1)}\sinh cx\cosh^{n+1} cx - \frac{n+2}{n+1}\int\cosh^{n+2}cx\,\mathrm{d}x \qquad\mbox{(pro }n<0\mbox{, }n\neq -1\mbox{)}\)
\(\int\frac{\mathrm{d}x}{\sinh cx} = \frac{1}{c} \ln\left|\tanh\frac{cx}{2}\right|\)
také: \(\int\frac{\mathrm{d}x}{\sinh cx} = \frac{1}{c} \ln\left|\frac{\cosh cx - 1}{\sinh cx}\right|\)
také: \(\int\frac{\mathrm{d}x}{\sinh cx} = \frac{1}{c} \ln\left|\frac{\sinh cx}{\cosh cx + 1}\right|\)
také: \(\int\frac{\mathrm{d}x}{\sinh cx} = \frac{1}{c} \ln\left|\frac{\cosh cx - 1}{\cosh cx + 1}\right|\)
\(\int\frac{\mathrm{d}x}{\cosh cx} = \frac{2}{c} \arctan e^{cx}\)
\(\int\frac{\mathrm{d}x}{\sinh^n cx} = \frac{\cosh cx}{c(n-1)\sinh^{n-1} cx}-\frac{n-2}{n-1}\int\frac{\mathrm{d}x}{\sinh^{n-2} cx} \qquad\mbox{(pro }n\neq 1\mbox{)}\)
\(\int\frac{\mathrm{d}x}{\cosh^n cx} = \frac{\sinh cx}{c(n-1)\cosh^{n-1} cx}+\frac{n-2}{n-1}\int\frac{\mathrm{d}x}{\cosh^{n-2} cx} \qquad\mbox{(pro }n\neq 1\mbox{)}\)
\(\int\frac{\cosh^n cx}{\sinh^m cx} \mathrm{d}x = \frac{\cosh^{n-1} cx}{c(n-m)\sinh^{m-1} cx} + \frac{n-1}{n-m}\int\frac{\cosh^{n-2} cx}{\sinh^m cx} \mathrm{d}x \qquad\mbox{(pro }m\neq n\mbox{)}\)
také: \(\int\frac{\cosh^n cx}{\sinh^m cx} \mathrm{d}x = -\frac{\cosh^{n+1} cx}{c(m-1)\sinh^{m-1} cx} + \frac{n-m+2}{m-1}\int\frac{\cosh^n cx}{\sinh^{m-2} cx} \mathrm{d}x \qquad\mbox{(pro }m\neq 1\mbox{)}\)
také: \(\int\frac{\cosh^n cx}{\sinh^m cx} \mathrm{d}x = -\frac{\cosh^{n-1} cx}{c(m-1)\sinh^{m-1} cx} + \frac{n-1}{m-1}\int\frac{\cosh^{n-2} cx}{\sinh^{m-2} cx} \mathrm{d}x \qquad\mbox{(pro }m\neq 1\mbox{)}\)
\(\int\frac{\sinh^m cx}{\cosh^n cx} \mathrm{d}x = \frac{\sinh^{m-1} cx}{c(m-n)\cosh^{n-1} cx} + \frac{m-1}{m-n}\int\frac{\sinh^{m-2} cx}{\cosh^n cx} \mathrm{d}x \qquad\mbox{(pro }m\neq n\mbox{)}\)
také: \(\int\frac{\sinh^m cx}{\cosh^n cx} \mathrm{d}x = \frac{\sinh^{m+1} cx}{c(n-1)\cosh^{n-1} cx} + \frac{m-n+2}{n-1}\int\frac{\sinh^m cx}{\cosh^{n-2} cx} \mathrm{d}x \qquad\mbox{(pro }n\neq 1\mbox{)}\)
také: \(\int\frac{\sinh^m cx}{\cosh^n cx} \mathrm{d}x = -\frac{\sinh^{m-1} cx}{c(n-1)\cosh^{n-1} cx} + \frac{m-1}{n-1}\int\frac{\sinh^{m -2} cx}{\cosh^{n-2} cx} \mathrm{d}x \qquad\mbox{(pro }n\neq 1\mbox{)}\)
\(\int x\sinh cx\,\mathrm{d}x = \frac{1}{c} x\cosh cx - \frac{1}{c^2}\sinh cx\)
\(\int x\cosh cx\,\mathrm{d}x = \frac{1}{c} x\sinh cx - \frac{1}{c^2}\cosh cx\)
\(\int \tanh cx\,\mathrm{d}x = \frac{1}{c}\ln|\cosh cx|\)
\(\int \coth cx\,\mathrm{d}x = \frac{1}{c}\ln|\sinh cx|\)
\(\int \tanh^n cx\,\mathrm{d}x = -\frac{1}{c(n-1)}\tanh^{n-1} cx+\int\tanh^{n-2} cx\,\mathrm{d}x \qquad\mbox{(pro }n\neq 1\mbox{)}\)
\(\int \coth^n cx\,\mathrm{d}x = -\frac{1}{c(n-1)}\coth^{n-1} cx+\int\coth^{n-2} cx\,\mathrm{d}x \qquad\mbox{(pro }n\neq 1\mbox{)}\)
\(\int \sinh bx \sinh cx\,\mathrm{d}x = \frac{1}{b^2-c^2} (b\sinh cx \cosh bx - c\cosh cx \sinh bx) \qquad\mbox{(pro }b^2\neq c^2\mbox{)}\)
\(\int \cosh bx \cosh cx\,\mathrm{d}x = \frac{1}{b^2-c^2} (b\sinh bx \cosh cx - c\sinh cx \cosh bx) \qquad\mbox{(pro }b^2\neq c^2\mbox{)}\)
\(\int \cosh bx \sinh cx\,\mathrm{d}x = \frac{1}{b^2-c^2} (b\sinh bx \sinh cx - c\cosh bx \cosh cx) \qquad\mbox{(pro }b^2\neq c^2\mbox{)}\)
\(\int \sinh (ax+b)\sin (cx+d)\,\mathrm{d}x = \frac{a}{a^2+c^2}\cosh(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\sinh(ax+b)\cos(cx+d)\)
\(\int \sinh (ax+b)\cos (cx+d)\,\mathrm{d}x = \frac{a}{a^2+c^2}\cosh(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\sinh(ax+b)\sin(cx+d)\)
\(\int \cosh (ax+b)\sin (cx+d)\,\mathrm{d}x = \frac{a}{a^2+c^2}\sinh(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\cosh(ax+b)\cos(cx+d)\)
\(\int \cosh (ax+b)\cos (cx+d)\,\mathrm{d}x = \frac{a}{a^2+c^2}\sinh(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\cosh(ax+b)\sin(cx+d)\)

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